9. AFTERGLOW

It is generally believed that the observed afterglow results from
slowing down of a relativistic shell on the external ISM. The
afterglow is produced, in this case, by an external shock. A second
alternative is of "continuous emission". The "inner engine" that
powers the GRB continues to emit energy for much longer duration with
a lower amplitude [36]
and may produce the earlier part
(first day or two in GRB970228 and GRB970508) of the afterglow. It is
most likely that both processes take place to some extent
[26].
We discuss in this section theoretical models for the
production of the afterglow focusing on the external shock model.

Within the external shock model there are several possible physical
assumptions that one can make. The "standard" model assumes
adiabatic hydrodynamics (energy losses are negligible and do not
influence the hydrodynamics), slow cooling (the electrons radiate a
small fraction of the energy that is generated by the shock) and
synchrotron emission
[17,
18,
21,
22,
23,
247,
47].
However there are other possibilities. First, the electrons' energy
might be radiated rapidly. In this case the radiation process is fast
and the observed flux is determined by the rate of energy generation
by the shock. If the electrons carry a significant fraction of the
total internal energy fast cooling will influence the hydrodynamics
which will not be adiabatic any more. In this case we have a radiative
solution [25,
24]
which differs in its basic scaling
laws from the adiabatic one. The different possibilities are summarized in
Table 9.1

We consider first a simple model for the slowing down of the shell.
In this model the slowing down is described by a series of
infinitesimal inelastic collisions between the shell and
infinitesimal external masses. We assume a homogeneous shell described
by its rest frame energy M (rest mass and thermal energy) and its
Lorentz factor
.
Initially,
E0 = M0c20.
The shell
collides with the surrounding matter. We denote the mass of the ISM
that has already collided with the shell by m(R). As the shell
propagates it sweeps up more ISM mass. Additional ISM mass elements,
dm, which are at rest collides inelastically with the shell.

Energy and momentum conservation yield:

(111)

and

(112)

where dE is the thermal energy produced in this collision. We
define as the
fraction of the shock generated thermal
energy (relative to the observer frame) that is radiated. The
incremental total mass satisfies:

(113)

These equations yields analytic relations between the Lorentz factor
and the total mass of the shell:

(114)

and between m(R) (and therefore R) and
.

(115)

These relations completely describe the hydrodynamical evolution of
the shell.

Two basic features can be seen directly from Eq. 116.
First, we can estimate the ISM mass m that should be swept to get
significant deceleration. Solving Eq. 116 with an upper limit
0
/ 2 and using
0
>> 1 we obtain the well known result: a mass
mM0 /
(20) is required to reach
=
0
/ 2. Apparently this result is independent
of the cooling parameter
.

A second simple result can be obtained in the limit that
0
>>
>> 1:

(116)

so that R-3/(2-). For
= 0 this
yields the well known adiabatic result:

For comparison with observations we have to calculate the observed
time that corresponds to different radii and Lorentz factors.
The well known formula

(119)

is valid only for emission along the line of sight
from a shell that propagates with a constant velocity. Sari
[253]
pointed out that as the shell decelerates this formula should be used
only in a differential sense:

(120)

Eq. 120 should be combined with the relation 117
or 118 and integrated to get the actual relation between
observed time and emission radius. For an adiabatic expansion, for
example, this yields: tobs = R /
162c [253].
Eq. 120 is valid only along the line of sight. The situation
is complicated further if we recall that the emission reaches the
observe from an angle of order
-1
around the line of sight.
Averaging on all angles yields another numerical factor
[254,
255,
256]
and altogether we get

(121)

where the value of the
numerical factor, cga, depends on the details of the
solution and it varies between ~ 3 and ~ 7.
Using Eqs. 121 and 117 or 118 we obtain the following relations between
R,
and t:

(122)

(123)

where L (3E /
4n mpc2)1/3 is the radius
where the external mass equals the mass of the shell.

One can proceed and use the relation between R and
and
tobs (Eqs. 122 and 123) to estimate the physical
conditions at the shocked material using Eqs. 44. Then one can
estimate the emitted radiation from this shock using Eqs.
56 and 57. However, before doing so we explore
the Blandford-McKee self similar solution
[238], which
describes more precisely the adiabatic expansion. This solution is
inhomogeneous with a well determined radial profile. The matter at
the front of the shell moves faster than the average speed. This
influences the estimates of the radiation emitted from the shell.

Blandford & McKee
[238]
discovered a self-similar solution
that describes the adiabatic slowing down of an extremely relativistic
shell propagating into the ISM. Using several simplifications and
some algebraic manipulations we rewrite the Blandford-McKee solution as
[253]:

(124)

where n(r, t), e(r, t) and
(r,
t) are, respectively, the
density, energy density and Lorentz factor of the material behind the
shock (not to be confused with the ISM density n) and
(t)
= (R(t)) is the Lorentz factor of material just
behind the shock. n(r, t) and e(r,
t) are measured in the fluid's rest frame while
(r,
t) is relative to an observer at rest. The
total energy in this adiabatic flow equals E =
E0, the initial energy. The scaling laws of
R(t) and
(t)
that follow from these profiles and from the condition that the total
energy in the flow equals E is:

(125)

The scalings 125 are consistent with the scalings 122
and 123 which were derived using conservation of energy and
momentum. They provide the exact numerical factor that cannot be
calculated by the simple analysis of
section 9.1.1. These equations can serve as a starting
point for a detailed radiation emission calculation and a comparison
with observations.

The Blandford-McKee solution is adiabatic and as such it does
not allow for any energy losses.
With some simplifying assumptions it is possible to derive a
self-similar radiative solution in which an arbitrary fraction of
the energy generated by the shock is radiated away
[257].